{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import sympy as sy\n",
    "sy.init_printing() "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# <font face=\"gotham\" color=\"purple\"> Linear Transformation </font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There many new terminologies in this chapter, however they are not entirely new to us."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let $V$ and $W$ be vector spaces. The mapping $T:\\ V\\rightarrow W$ is called a <font face=\"gotham\" color=\"red\">linear transformation</font> if an only if\n",
    "\n",
    "$$\n",
    "T(u+v)=T(u)+T(v)\\quad \\text{and} \\quad T(cu)=cT(u)\n",
    "$$\n",
    "\n",
    "for all $u,v\\in V$ and all $c\\in R$. If $T:\\ V\\rightarrow W$, then $T$ is called a <font face=\"gotham\" color=\"red\">linear operator</font>. For each $u\\in V$, the vector $w=T(u)$ is called the <font face=\"gotham\" color=\"red\">image</font> of $u$ under $T$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <font face=\"gotham\" color=\"purple\"> Parametric Function Plotting </font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We need one tool for illustrating the idea of linear transformation.\n",
    "\n",
    "We want to plot any line in vector space by an equation: <font face=\"gotham\" color=\"red\"> $p = p_0+tv$</font>. We need to know vector $p_0$ and $v$ to plot the line. \n",
    "\n",
    "For instance, $p_0 = (2, 6)$, $v=(5, 3)$ and $p = (x, y)$, subsitute them into our equation\n",
    "$$\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "x\\\\y\n",
    "\\end{matrix}\n",
    "\\right]=\\left[\n",
    "\\begin{matrix}\n",
    "2\\\\6\n",
    "\\end{matrix}\n",
    "\\right]+\n",
    "t\\left[\n",
    "\\begin{matrix}\n",
    "5\\\\3\n",
    "\\end{matrix}\n",
    "\\right]\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will create a plot to illustrate the linear transformation later."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "def paraEqPlot(p0, v0, p1, v1):\n",
    "    t = np.linspace(-5, 5)\n",
    "    ################### First Line ####################\n",
    "    fig, ax = plt.subplots(figsize = (10, 10))\n",
    "    x = p0[0,:] + v0[0,:]*t\n",
    "    y = p0[1,:] + v0[1,:]*t\n",
    "    ax.plot(x, y, lw = 3, color = 'red')\n",
    "    ax.grid(True)\n",
    "    ax.scatter(p0[0,:], p0[1,:], s = 150, ec = 'red', fc = 'black', zorder = 3)\n",
    "    \n",
    "    ################### First Line ####################\n",
    "    x = p1[0,:] + v1[0,:]*t\n",
    "    y = p1[1,:] + v1[1,:]*t\n",
    "    ax.plot(x, y, lw = 3, color = 'blue')\n",
    "    ax.grid(True)\n",
    "    ax.scatter(p1[0,:], p1[1,:], s = 150, ec = 'red', fc = 'black', zorder = 3)\n",
    "    \n",
    "    ax.spines['left'].set_position('zero')\n",
    "    ax.spines['bottom'].set_position('zero')\n",
    "\n",
    "    # Eliminate upper and right axes\n",
    "    ax.spines['right'].set_color('none')\n",
    "    ax.spines['top'].set_color('none')\n",
    "\n",
    "    # Show ticks in the left and lower axes only\n",
    "    ax.xaxis.set_ticks_position('bottom')\n",
    "    ax.yaxis.set_ticks_position('left')\n",
    "    \n",
    "    string =  '$(%.d, %.d)$' % (p0[0,:], p0[1,:])\n",
    "    ax.text(x= p0[0,:]+.5, y = p0[1,:], s = string, size = 14)\n",
    "    \n",
    "    string =  '$(%.d, %.d)$' % (p1[0,:], p1[1,:])\n",
    "    ax.text(x= p1[0,:]+.5, y = p1[1,:], s = string, size = 14)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <font face=\"gotham\" color=\"purple\"> A Simple Linear Transformation </font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we know the parametric functions in $\\mathbb{R}^2$, we can show how a linear transformation acturally works on a line."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's say, we perform linear transformation on a vector $(x, y)$, \n",
    "\n",
    "$$\n",
    "T\\left(\\left[\\matrix{x\\cr y}\\right]\\right)=\\pmatrix{3x-2y\\cr -2x+3y}\\\\\n",
    "$$\n",
    "and substitute the parametric function into the linear operator.\n",
    "\n",
    "$$\n",
    "T\\left(\\left[\\matrix{4+t\\cr 5+3t}\\right]\\right)=\\pmatrix{3(4+t)-2(5+3t)\\cr -2(4+t)+3(5+3t)}=\\left[\n",
    "\\begin{matrix}\n",
    "2-3t\\\\7+7t\n",
    "\\end{matrix}\n",
    "\\right]\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The red line is transformed into"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "p0 = np.array([[4],[5]])\n",
    "v0 = np.array([[1],[3]])\n",
    "p1 = np.array([[2],[7]])\n",
    "v1 = np.array([[-3],[7]])\n",
    "paraEqPlot(p0,v0,p1, v1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <font face=\"gotham\" color=\"purple\"> Visualization of Change of Basis </font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Change of basis is also a kind of linear transformation. Let's create a grid."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "u1, u2 = np.linspace(-5, 5, 10), np.linspace(-5, 5, 10)\n",
    "U1, U2 = np.meshgrid(u1, u2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We plot each row of $U2$ again each row of $U1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 720x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize = (10, 10))\n",
    "ax.plot(U1,U2, color = 'black') \n",
    "ax.plot(U1.T,U2.T, color = 'black') \n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let $A$ and $B$ be two bases in $\\mathbb{R}^3$\n",
    " \n",
    "$$\n",
    "A=\\left\\{\\left[\\matrix{2\\cr 1}\\right],\\ \\left[\\matrix{1\\cr 1}\\right]\\right\\}\\\\\n",
    "B=\\left\\{\\left[\\matrix{3\\cr 2}\\right],\\ \\left[\\matrix{0\\cr -1}\\right]\\right\\}\\\\\n",
    "$$\n",
    "\n",
    "If we want to use basis $A$ to represent $B$, we can construct an augmented matrix like we did before.\n",
    "\n",
    "$$\n",
    "[A|B]=\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "2 & 1 & 3 & 0\\\\\n",
    "1 & 1 & 2 & -1\n",
    "\\end{matrix}\n",
    "\\right]\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}1 & 0 & 1 & 1\\\\0 & 1 & 1 & -2\\end{matrix}\\right], \\  \\left( 0, \\  1\\right)\\right)$"
      ],
      "text/plain": [
       "⎛⎡1  0  1  1 ⎤        ⎞\n",
       "⎜⎢           ⎥, (0, 1)⎟\n",
       "⎝⎣0  1  1  -2⎦        ⎠"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AB = sy.Matrix([[2,1,3,0],[1,1,2,-1]]); AB.rref()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We find the transition matrix $P_{A\\leftarrow B}$\n",
    "$$\n",
    "[A|B]=[I|P_{A\\leftarrow B}]\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can write\n",
    "\n",
    "$$\n",
    "\\big[x\\big]_A = P_{A\\leftarrow B}\\big[u\\big]_B\\\\\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "x_1\\\\x_2\n",
    "\\end{matrix}\n",
    "\\right]\n",
    "=\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "1 & 1\\\\1 & -2\n",
    "\\end{matrix}\n",
    "\\right]\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "u_1\\\\u_2\n",
    "\\end{matrix}\n",
    "\\right]\\\\\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Therefore\n",
    "$$\n",
    "x_1 = u_1+u_2\\\\\n",
    "x_2 = u_1 - 2u_2\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's plot original and transformed coordinates together."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "u1, u2 = np.linspace(-10, 10, 21), np.linspace(-10, 10, 21)\n",
    "U1, U2 = np.meshgrid(u1, u2)\n",
    "\n",
    "fig, ax = plt.subplots(figsize = (10, 10))\n",
    "ax.plot(U1,U2, color = 'black', lw = 1) \n",
    "ax.plot(U1.T,U2.T, color = 'black', lw = 1) \n",
    "\n",
    "X1 = U1 +U2\n",
    "X2 = U1 - 2*U2\n",
    "ax.plot(X1,X2, color = 'red', ls = '--') \n",
    "ax.plot(X1.T,X2.T, color = 'red', ls = '--') \n",
    "\n",
    "ax.arrow(0, 0, 1, 1, color = 'blue', width = .07, \n",
    "     length_includes_head = True,\n",
    "     head_width = .2, # default: 3*width\n",
    "     head_length = .3, zorder = 4,\n",
    "     overhang = .4)\n",
    "\n",
    "ax.arrow(0, 0, 1, -2, color = 'blue', width = .07, \n",
    "     length_includes_head = True,\n",
    "     head_width = .2, # default: 3*width\n",
    "     head_length = .3,zorder = 4,\n",
    "     overhang = .4)\n",
    "\n",
    "ax.text(0.1,0.1,'$(0, 0)$',size = 14)\n",
    "ax.scatter(0,0,s = 120, zorder = 5, ec = 'red', fc = 'black')\n",
    "\n",
    "ax.axis([-4, 4, -5, 5])\n",
    "plt.show()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.4"
  },
  "toc": {
   "base_numbering": 1,
   "nav_menu": {},
   "number_sections": true,
   "sideBar": true,
   "skip_h1_title": false,
   "title_cell": "Table of Contents",
   "title_sidebar": "Contents",
   "toc_cell": false,
   "toc_position": {},
   "toc_section_display": true,
   "toc_window_display": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}
